DOCSLIB.ORG

Electronic Density-Matrix Algorithm for Nonadiabatic Couplings in Molecular Dynamics Simulations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Electronic Density-Matrix Algorithm for Nonadiabatic Couplings in Molecular Dynamics Simulations Electronic Density-Matrix Algorithm for Nonadiabatic Couplings in Molecular Dynamics Simulations M. TOMMASINI,1 V. CHERNYAK,2 S. MUKAMEL2 1Dipartimento di Chimica Industriale e Ingegneria Chimica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy 2Department of Chemistry, University of Rochester, P. O. RC Box 270216 Rochester, NY 14627-0216 Received 25 February 2001; accepted 2 March 2001 ABSTRACT: Closed expressions for nonadiabatic couplings are derived using the collective electronic oscillators (CEO) algorithm based on the time-dependent Hartree–Fock equations. Analytic derivatives allow the calculation of transition density matrices and potential surfaces at arbitrary nuclear geometries using a molecular dynamics trajectory that only requires a CEO calculation at a single configuration. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem 00: 1–14, 2001 Key words: trix between ground and excited states, is computed Introduction instead [5]. These matrices further represent col- lective oscillators or electronic normal modes that he simulation of photoinduced processes in describe the time evolution of the density matrix of T molecules requires the computation of elec- the system under the effects of an external perturba- tronic potential surfaces and nonadiabatic cou- tion. The CEO eigenvalue equation can be written as plings between them [1, 2]. The collective electronic Lξk = kξk.(1) oscillators approach (CEO) provides an inexpensive algorithm for computing adiabatic potential sur- L is the effective TDHF Liouville operator for the faces [3, 4]. This constitutes a major computational molecule, whose spectrum describes the electronic saving since a quantity which carries much less in- excitations of the system. The atomic orbital rep- formation than the many electron wavefunction of resentation of the oscillators ξk provides a clear the excited states, namely the transition density ma- characterization of molecular excitations in real space [6]. The eigenvectors ξk are normalized ac- 1 = ¯ † = ¯ Correspondence to: S. Mukamel; e-mail: mukamel@chem. cording to: (ξk, ξk) Tr(ρ[ξk , ξk]) 1, where ρ is rochester.edu. Contract grant sponsor: National Science Foundation and the 1The scalar product introduced here to normalize the eigen- Petroleum Research Fund, American Chemical Society. vectors ξk is discussed in Appendix A. International Journal of Quantum Chemistry, Vol. 00, 1–14 (2001) © 2001 John Wiley & Sons, Inc. TOMMASINI, CHERNYAK, AND MUKAMEL the Hartree–Fock ground state density matrix. The has been rewritten in terms of the ground state plus x sum of the excitation energy k(R)andtheground the excitation energy εk = ε0 + k.Thematrixh ˆx state energy ε0(R) at different nuclear configura- represents the operator h in the chosen basis set tions R gives the excited state adiabatic potential ˆx = x † h nmhnmcncm. The expression for NAC involving surface εk(R). These quantities are sufficient to de- the ground state is even simpler and only requires scribe the nuclear dynamics of a molecule in regions † the knowledge of (ρi0)nm =i|cncm|0≡(ξi)nm and of the nuclear space far away from avoided cross- † † (ρ0i)nm =0|c cm|i≡(ξ )nm ings or conical intersections. However, many in- n i teresting photoinduced processes involve the tran- ξ x Tr(ξ †hx) x = Tr( ih ) x = i sition between different potential surfaces in the Ai0 ; A0i .(5) i i vicinity of an avoided crossing or conical intersec- The matrices {ξ } are obtained from the collective tion [7]. An example of such a process is the cis to i electronic oscillators algorithm [6, 12]. The transi- trans photoisomerization of rhodopsin which is the tion density matrices between excited states, ρ ,can elementary process in vision [8 – 10]. Under these ik be also expressed in terms of the CEO [13]. To use circumstances it is crucial to include the nonadia- ˆx batic couplings between different potential surfaces Eqs. (4) and (5) we need to express the operator h in order to properly describe the joint electronic and in terms of the derivative of the matrix elements of the molecular Hamiltonian. In Appendix C we nuclear dynamics, and to predict the branching ra- ˆ tios among different reaction pathways. show that the matrix elements of the operator hx The aim of this work is to derive closed ex- canbeexpressedinanatomicorbitalrepresenta- pressions for the nonadiabatic couplings (NAC) tion as between any pair of electronic states in the CEO hx = tx + Vxρ¯ (6) framework. The close connection between the NAC and the derivatives of the density matrices further where tx is the derivative of the matrix elements provides the formal solution of the equations pro- for the one-electron part of the Hamiltonian and Vx posed in [3] in terms of CEO eigenvectors and NAC. is the derivative of the electron–electron interaction ˆ The first-order nonadiabatic couplings are con- operator. The operator hx represents the vibronic veniently defined in terms of a vector potential in coupling of the molecular Hamiltonian and it is the nuclear space [11] source of the nonadiabatic coupling. These results will be discussed in the coming sections. The sec- ∂ Ax ≡− i k (2) ond section presents the derivatives of the ground ik ∂x state density matrix and of the CEO transition den- where i| and k| are two adiabatic electronic states sity matrices. In the third section we present the and x is a generic nuclear coordinate. In Appendix B equations for classical molecular dynamics on ex- we use perturbation theory to express the NAC in cited state hypersurfaces using the CEO/TDHF ap- the form proach. An analytic expression for the derivatives |ˆx| of the density matrices is also derived. In the fourth x = i h k = Aik ; i k (3) section we discuss some general properties of nona- εi − εk diabatic couplings resulting from the translational ˆx where h is the derivative of the molecular Hamil- symmetry of the electronic Hamiltonian when ex- ˆ 2 tonian H with respect to a nuclear coordinate x. pressed using atomic orbitals centered on the nuclei. This expression is convenient for numerical compu- Similar properties are derived also for the density tations. Introducing a basis set, with the correspond- matrices. In the fifth section we discuss the role † ing fermionic creation and annihilation operators cn of nonadiabatic couplings in determining a contri- and cn,wehave bution to vibrational infrared absorption intensi- ρ x ties. Finally, the Appendixes present details of the x = Tr( ikh ) = Aik ; i k.(4)derivations and the CEO algorithm. i − k =| † | (ρik)nm i cncm k denotes the one-electron tran- sition density matrix and the excited state energy Analytic Derivatives 2Throughout this work we use the short-hand notation and denote the derivative with respect to a nuclear coordinate x by x By solving the Hartree–Fock equation for the superscript. density matrix at various nuclear configurations 2 VOL. 00, NO. 0 NONADIABATIC COUPLINGS and the CEO eigenvalue problem [Eq. (1)] it is possi- x = ξ , Lxξ ; i i i bletoconstructtheentireadiabaticpotentialenergy Lxζ = F(ρ¯)x, ζ + Vxζ , ρ¯ + Vζ , ρ¯x . surfaces. The nonadiabatic couplings at each nu- clear configuration can then be computed from the Here and in the remainder of this paper ζ denotes a knowledge of the CEO transition matrices and the matrix representing an arbitrary single electron op- transition energies, using Eqs. (4) and (5). A differ- erator: Eq. (9) shows the action of superoperators in ent strategy for achieving the same goal is to use Liouville space. The scalar product (ζ1, ζ2)inEq.(9) the analytic derivatives of all quantities with re- is defined as spect to the nuclear coordinates. It is then possible † (ζ , ζ ) = (ζ |ζ ) = Tr ρ¯[ζ , ζ ] . (10) in principle to carry out the full Hartree–Fock/CEO 1 2 1 2 1 2 calculation only once, at some chosen initial config- With respect to this scalar product the Liouville op- uration. The analytic derivatives then allow us to erator Lζ = [F(ρ¯), ζ ]+[Vζ , ρ¯]ofEq.(1)isHermitian: x compute the quantities of interest at all configura- (Lη1, η2) = (η1, Lη2) (see Appendix A). F(ρ¯) is the tions by running a trajectory; we can thus generate derivative of the Fock matrix corresponding to the the density matrices on the fly while avoiding a new ground state density matrix ρ¯ (see Appendix H). Hartree–Fock SCF or a new CEO computation in the displaced nuclear configuration. Such strategy is reminiscent of Car–Parrinello simulations [14]. In Classical Nuclear this section we provide closed expressions for these Dynamics Simulations derivatives. Numerical considerations should deter- mine which strategy is to be preferred: repeating the Many processes and spectroscopic properties CEO or using the derivatives. Making use of these may be adequately simulated by running classical results for the NAC it is possible to compute the trajectories on excited state surfaces. However, the derivative of the ground state density matrix and classical description breaks down at curve crossing the derivative of the CEO eigenvectors. points where a quantum wave packet calculation is A general relationship exists between the deriv- required [7]. In [3] the Hartree–Fock equation for ative of the density matrices and the NAC (see Ap- the ground state and the CEO algorithm for the pendix D) TDHF were differentiated with respect to nuclear coordinates in order to get closed equations that can ρx = Ax ρ − ρ Ax .(7) ij ik kj ik kj describe the molecular dynamics on excited state k = i k = j hypersurfaces. We recast here the equations of [3] In particular we have for the ground state density using a somewhat different notation and adopting matrix a different point of view which underscores the ¯x = x − x † interesting role of the nonadiabatic couplings. We ρ A0jξj Aj0ξj .(8) j>0 consider both electronic and nuclear degrees of free- dom and write equations of motion for the variables Equation (8) shows that the derivative of the ground of interest along the classical trajectory of the nuclei.
Load more

Recommended publications
  • Arxiv:1809.04476V2 [Physics.Chem-Ph] 18 Oct 2018 (Sub)States
    Quantum System Partitioning at the Single-Particle Level Quantum System Partitioning at the Single-Particle Level Adrian H. M¨uhlbach and Markus Reihera) ETH Z¨urich, Laboratorium f¨ur Physikalische Chemie, Vladimir-Prelog-Weg 2, CH-8093 Z¨urich,Switzerland (Dated: 17 October 2018) We discuss the partitioning of a quantum system by subsystem separation through unitary block- diagonalization (SSUB) applied to a Fock operator. For a one-particle Hilbert space, this separation can be formulated in a very general way. Therefore, it can be applied to very different partitionings ranging from those driven by features in the molecular structure (such as a solute surrounded by solvent molecules or an active site in an enzyme) to those that aim at an orbital separation (such as core-valence separation). Our framework embraces recent developments of Manby and Miller as well as older ones of Huzinaga and Cantu. Projector-based embedding is simplified and accelerated by SSUB. Moreover, it directly relates to decoupling approaches for relativistic four-component many-electron theory. For a Fock operator based on the Dirac one-electron Hamiltonian, one would like to separate the so-called positronic (negative-energy) states from the electronic bound and continuum states. The exact two-component (X2C) approach developed for this purpose becomes a special case of the general SSUB framework and may therefore be viewed as a system- environment decoupling approach. Moreover, for SSUB there exists no restriction with respect to the number of subsystems that are generated | in the limit, decoupling of all single-particle states is recovered, which represents exact diagonalization of the problem.
    [Show full text]
  • A New Scalable Parallel Algorithm for Fock Matrix Construction
    A New Scalable Parallel Algorithm for Fock Matrix Construction Xing Liu Aftab Patel Edmond Chow School of Computational Science and Engineering College of Computing, Georgia Institute of Technology Atlanta, Georgia, 30332, USA [email protected], [email protected], [email protected] Abstract—Hartree-Fock (HF) or self-consistent field (SCF) ber of independent tasks and using a centralized dynamic calculations are widely used in quantum chemistry, and are scheduler for scheduling them on the available processes. the starting point for accurate electronic correlation methods. Communication costs are reduced by defining tasks that Existing algorithms and software, however, may fail to scale for large numbers of cores of a distributed machine, particularly in increase data reuse, which reduces communication volume, the simulation of moderately-sized molecules. In existing codes, and consequently communication cost. This approach suffers HF calculations are divided into tasks. Fine-grained tasks are from a few problems. First, because scheduling is completely better for load balance, but coarse-grained tasks require less dynamic, the mapping of tasks to processes is not known communication. In this paper, we present a new parallelization in advance, and data reuse suffers. Second, the centralized of HF calculations that addresses this trade-off: we use fine- grained tasks to balance the computation among large numbers dynamic scheduler itself could become a bottleneck when of cores, but we also use a scheme to assign tasks to processes to scaling up to a large system [14]. Third, coarse tasks reduce communication. We specifically focus on the distributed which reduce communication volume may lead to poor load construction of the Fock matrix arising in the HF algorithm, balance when the ratio of tasks to processes is close to one.
    [Show full text]
  • Domain Decomposition and Electronic Structure Computations: a Promising Approach
    Domain Decomposition and Electronic Structure Computations: A Promising Approach G. Bencteux1,4, M. Barrault1, E. Canc`es2,4, W. W. Hager3, and C. Le Bris2,4 1 EDF R&D, 1 avenue du G´en´eral de Gaulle, 92141 Clamart Cedex, France {guy.bencteux,maxime.barrault}@edf.fr 2 CERMICS, Ecole´ Nationale des Ponts et Chauss´ees, 6 & 8, avenue Blaise Pascal, Cit´eDescartes, 77455 Marne-La-Vall´ee Cedex 2, France, {cances,lebris}@cermics.enpc.fr 3 Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, USA, [email protected] 4 INRIA Rocquencourt, MICMAC project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, FRANCE Summary. We describe a domain decomposition approach applied to the spe- cific context of electronic structure calculations. The approach has been introduced in [BCH06]. We survey here the computational context, and explain the peculiari- ties of the approach as compared to problems of seemingly the same type in other engineering sciences. Improvements of the original approach presented in [BCH06], including algorithmic refinements and effective parallel implementation, are included here. Test cases supporting the interest of the method are also reported. It is our pleasure and an honor to dedicate this contribution to Olivier Pironneau, on the occasion of his sixtieth birthday. With admiration, respect and friendship. 1 Introduction and Motivation 1.1 General Context Numerical simulation is nowadays an ubiquituous tool in materials science, chemistry and biology. Design of new materials, irradiation induced damage, drug design, protein folding are instances of applications of numerical sim- ulation. For convenience we now briefly present the context of the specific computational problem under consideration in the present article.
    [Show full text]
  • Matrix Algebra for Quantum Chemistry
    Matrix Algebra for Quantum Chemistry EMANUEL H. RUBENSSON Doctoral Thesis in Theoretical Chemistry Stockholm, Sweden 2008 Matrix Algebra for Quantum Chemistry Doctoral Thesis c Emanuel Härold Rubensson, 2008 TRITA-BIO-Report 2008:23 ISBN 978-91-7415-160-2 ISSN 1654-2312 Printed by Universitetsservice US AB, Stockholm, Sweden 2008 Typeset in LATEX by the author. Abstract This thesis concerns methods of reduced complexity for electronic structure calculations. When quantum chemistry methods are applied to large systems, it is important to optimally use computer resources and only store data and perform operations that contribute to the overall accuracy. At the same time, precarious approximations could jeopardize the reliability of the whole calcu- lation. In this thesis, the selfconsistent eld method is seen as a sequence of rotations of the occupied subspace. Errors coming from computational ap- proximations are characterized as erroneous rotations of this subspace. This viewpoint is optimal in the sense that the occupied subspace uniquely denes the electron density. Errors should be measured by their impact on the over- all accuracy instead of by their constituent parts. With this point of view, a mathematical framework for control of errors in HartreeFock/KohnSham calculations is proposed. A unifying framework is of particular importance when computational approximations are introduced to eciently handle large systems. An important operation in HartreeFock/KohnSham calculations is the calculation of the density matrix for a given Fock/KohnSham matrix. In this thesis, density matrix purication is used to compute the density matrix with time and memory usage increasing only linearly with system size. The forward error of purication is analyzed and schemes to control the forward error are proposed.
    [Show full text]
  • Problem Set #1 Solutions
    PROBLEM SET #2 SOLUTIONS by Robert A. DiStasio Jr. Q1. 1-particle density matrices and idempotency. (a) A matrix M is said to be idempotent if M 2 = M . Show from the basic definition that the HF density matrix is idempotent when expressed in an orthonormal basis. An element of the HF density matrix is given as (neglecting the factor of two for the restricted closed-shell HF density matrix): * PCCμν = ∑ μi ν i . (1) i In matrix form, (1) is simply equivalent to † PCC= occ occ (2) where Cocc is the occupied block of the MO coefficient matrix. To reformulate the nonorthogonal Roothaan-Hall equations FC= SCε (3) into the preferred eigenvalue problem ~~ ~ FC= Cε (4) the AO basis must be orthogonalized. There are several ways to orthogonalize the AO basis, but I will only discuss symmetric orthogonalization, as this is arguably the most often used procedure in computational quantum chemistry. In the symmetric orthogonalization procedure, the MO coefficient matrix in (3) is recast as ~ + 1 − 1 ~ CSCCSC=2 ⇔ = 2 (5) which can be substituted into (3) yielding the eigenvalue form of the Roothaan-Hall equations in (4) after the following algebraic manipulation: FC = SCε − 1 ~ − 1 ~ FSCSSC( 2 )(= 2 )ε − 1 − 1 ~ −1 − 1 ~ SFSCSSSC2 ( 2 )(= 2 2 )ε . (6) −1 − 1 ~ −1 − 1 ~ ()S2 FS2 C= () S2 SS2 Cε ~~ ~ FC = Cε ~ ~ −1 − 1 −1 − 1 In (6), the dressed Fock matrix, F , was clearly defined as F≡ S2 FS 2 and S2 SS2 = S 0 =1 was used. In an orthonormal basis, therefore, the density matrix of (2) takes on the following form: ~ ~† PCC= occ occ .
    [Show full text]
  • Relativistic Cholesky-Decomposed Density Matrix
    Relativistic Cholesky-decomposed density matrix MP2 Benjamin Helmich-Paris,1, 2, a) Michal Repisky,3 and Lucas Visscher2 1)Max-Planck-Institut f¨urKohlenforschung, Kaiser-Wilhelm-Platz 1, D-45470 M¨ulheiman der Ruhr 2)Section of Theoretical Chemistry, Vrije Universiteit Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands 3)Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, UiT The Arctic University of Norway, N-9037 Tromø Norway (Dated: 15 October 2018) In the present article, we introduce the relativistic Cholesky-decomposed density (CDD) matrix second-order Møller–Plesset perturbation theory (MP2) energies. The working equations are formulated in terms of the usual intermediates of MP2 when employing the resolution-of-the-identity approximation (RI) for two-electron inte- grals. Those intermediates are obtained by substituting the occupied and vir- tual quaternion pseudo-density matrices of our previously proposed two-component atomic orbital-based MP2 (J. Chem. Phys. 145, 014107 (2016)) by the corresponding pivoted quaternion Cholesky factors. While working within the Kramers-restricted formalism, we obtain a formal spin-orbit overhead of 16 and 28 for the Coulomb and exchange contribution to the 2C MP2 correlation energy, respectively, compared to a non-relativistic (NR) spin-free CDD-MP2 implementation. This compact quaternion formulation could also be easily explored in any other algorithm to compute the 2C MP2 energy. The quaternion Cholesky factors become sparse for large molecules and, with a block-wise screening, block sparse-matrix multiplication algorithm, we observed an effective quadratic scaling of the total wall time for heavy-element con- taining linear molecules with increasing system size.
    [Show full text]
  • Lecture 16 Hartree Fock Implementation
    Quantum Chemistry Lecture 16 Implementation of Hartree-Fock Theory NC State University The linear combination of atomic orbitals (LCAO) Calculations of the energy and properties of molecules requires hydrogen-like wave functions on each of the nuclei. The Hartree-Fock method begins with assumption That molecular orbitals can be formed as a linear combination of atomic orbitals. The basis functions fm are hydrogen-like atomic orbitals that have been optimized by a variational procedure. The HF procedure is a variational procedure to minimize the coefficients Cmi. Note that we use the index m for atomic orbitals and i or j for molecular orbitals. Common types of atomic orbitals Slater-type orbitals (STOs) The STOs are like hydrogen atom wave functions. The problem with STOs arises in multicenter integrals. The Coulomb and exchange integrals involve electrons on different nuclei and so the distance r has a different origin. Gaussian-type orbitals (GTOs) Gaussian orbitals can be used to mimic the shape of exponentials, i.e. the form of the solutions for the hydrogen atom. Multicenter Gaussian integrals can be solved analytically. STOs vs GTOs GTOs are mathematically easy to work with STOs vs GTOs GTOs are mathematically easy to work with, but the shape of a Gaussian is not that similar to that of an exponential. STOs vs GTOs Therefore, linear combinations of Gaussians are used to imitate the shape of an exponential. Shown is a representation of the 3-Gaussian model of a STO. Double-zeta basis sets Since the remaining atoms have a different exponential dependence than hydrogen it is often convenient to include more parameters.
    [Show full text]
  • Fock Matrix Construction for Large Systems
    Fock Matrix Construction for Large Systems Elias Rudberg Theoretical Chemistry Royal Institute of Technology Stockholm 2006 Fock Matrix Construction for Large Systems Licentiate thesis c Elias Rudberg, 2006 ISBN 91-7178-535-3 ISBN 978-91-7178-535-0 Printed by Universitetsservice US AB, Stockholm, Sweden, 2006 Typeset in LATEX by the author Abstract This licentiate thesis deals with quantum chemistry methods for large systems. In particu- lar, the thesis focuses on the efficient construction of the Coulomb and exchange matrices which are important parts of the Fock matrix in Hartree–Fock calculations. The meth- ods described are also applicable in Kohn–Sham Density Functional Theory calculations, where the Coulomb and exchange matrices are parts of the Kohn–Sham matrix. Screening techniques for reducing the computational complexity of both Coulomb and exchange com- putations are discussed, as well as the fast multipole method, used for efficient computation of the Coulomb matrix. The thesis also discusses how sparsity in the matrices occurring in Hartree–Fock and Kohn– Sham Density Functional Theory calculations can be used to achieve more efficient storage of matrices as well as more efficient operations on them. As an example of a possible type of application, the thesis includes a theoretical study of Heisenberg exchange constants, using unrestricted Kohn–Sham Density Functional Theory calculations. iii Preface The work presented in this thesis has been carried out at the Department of Theoretical Chemistry, Royal Institute of Technology, Stockholm, Sweden. List of papers included in the thesis Paper 1 Efficient implementation of the fast multipole method, Elias Rudberg and Pawe l Sa lek,J.
    [Show full text]
  • Theoretical Studies of Charge Recombination Reactions
    Theoretical studies of charge recombination reactions Sifiso Musa Nkambule Akademisk avhandling f¨oravl¨aggandeav licentiatexamen vid Stockholms universitet, Fysikum. January, 2015. Abstract This thesis is based on theoretical studies that have been done for low-energy reactions involving small molecular systems. It is mainly focusing on mutual neutralization of oppositely charged ions and dissociative recombination of molecular ions. Both reactions involve highly excited electronic states that are coupled to each other. Employing electronic structure meth- ods, the electronic states relevant to the reactions are computed. It is necessary to go beyond the Born-Oppenheimer approximation and include non-adiabatic effects. This non-adiabatic behaviour plays a crucial role in driving the reactions. Schemes on how to go about and include these coupling elements are discussed in the thesis, which include computations of non-adiabatic and electronic couplings of the resonant states to the ionisation continuum. The nuclear dynamics are studied either semi-classically, using the Landau-Zener method or quantum mechanically, employing the time-independent and time-dependant Schr¨odinger + equations. Reactions studied here are mutual neutralization in collision of H + H−, where both the semi-classical and quantum mechanical methods are employed. Total and differential cross sections are computed for all hydrogen isotopes. We also perform quantum mechanical + studies of mutual neutralization in the collision of He + H−. For this system not only the non-adiabatic couplings among the neutral states have to be considered, but also the electronic coupling to the ionization continuum. Worth noting in the two systems is that reactants and the final products belong to different electronic states, which a coupled together, hence for the reaction to take place there must be non-adiabatic transition.
    [Show full text]
  • Quaternion Symmetry in Relativistic Molecular Calculations: the Dirac–Hartree–Fock Method T
    Quaternion symmetry in relativistic molecular calculations: The Dirac–Hartree–Fock method T. Saue, H. Jensen To cite this version: T. Saue, H. Jensen. Quaternion symmetry in relativistic molecular calculations: The Dirac–Hartree– Fock method. Journal of Chemical Physics, American Institute of Physics, 1999, 111 (14), pp.6211- 6222. 10.1063/1.479958. hal-03327446 HAL Id: hal-03327446 https://hal.archives-ouvertes.fr/hal-03327446 Submitted on 27 Aug 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL OF CHEMICAL PHYSICS VOLUME 111, NUMBER 14 8 OCTOBER 1999 Quaternion symmetry in relativistic molecular calculations: The Dirac–Hartree–Fock method T. Sauea) and H. J. Aa Jensen Department of Chemistry, University of Southern Denmark—Main campus: Odense University, DK-5230 Odense M, Denmark ͑Received 5 April 1999; accepted 9 July 1999͒ A symmetry scheme based on the irreducible corepresentations of the full symmetry group of a molecular system is presented for use in relativistic calculations. Consideration of time-reversal symmetry leads to a reformulation of the Dirac–Hartree–Fock equations in terms of quaternion algebra. Further symmetry reductions due to molecular point group symmetry are then manifested by a descent to complex or real algebra.
    [Show full text]
  • Hartree-Fock Theory
    Hartree-Fock theory Morten Hjorth-Jensen1 National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA1 May 16-20 2016 c 2013-2016, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license Why Hartree-Fock? Hartree-Fock (HF) theory is an algorithm for finding an approximative expression for the ground state of a given Hamiltonian. The basic ingredients are I Define a single-particle basis f αg so that HF h^ α = "α α with the Hartree-Fock Hamiltonian defined as HF HF h^ = t^+u ^ext +u ^ HF I The term u^ is a single-particle potential to be determined by the HF algorithm. HF I The HF algorithm means to choose u^ in order to have HF hH^i = E = hΦ0jH^jΦ0i that is to find a local minimum with a Slater determinant Φ0 being the ansatz for the ground state. HF I The variational principle ensures that E ≥ E0, with E0 the exact ground state energy. Why Hartree-Fock? We will show that the Hartree-Fock Hamiltonian h^HF equals our definition of the operator f^ discussed in connection with the new definition of the normal-ordered Hamiltonian (see later lectures), that is we have, for a specific matrix element HF X hpjh^ jqi = hpjf^jqi = hpjt^+u ^extjqi + hpijV^jqiiAS ; i≤F meaning that HF X hpju^ jqi = hpijV^jqiiAS : i≤F The so-called Hartree-Fock potential u^HF brings an explicit medium dependence due to the summation over all single-particle states below the Fermi level F .
    [Show full text]
  • Restricted Closed Shell Hartree Fock Roothaan Matrix Method Applied to Helium Atom Using Mathematica Introduction the Hartree Fo
    European J of Physics Education Volume 5 Issue 1 2014 Acosta, Tapia, Cab Restricted Closed Shell Hartree Fock Roothaan Matrix Method Applied to Helium Atom using Mathematica César R. Acosta*1 J. Alejandro Tapia1 César Cab1 1Av. Industrias no Contaminantes por anillo periférico Norte S/N Apdo Postal 150 Cordemex Physics Engineering Department Universidad Autónoma de Yucatán *[email protected] (Received: 20.12. 2013, Accepted: 13.02.2014) Abstract Slater type orbitals were used to construct the overlap and the Hamiltonian core matrices; we also found the values of the bi-electron repulsion integrals. The Hartree Fock Roothaan approximation process starts with setting an initial guess value for the elements of the density matrix; with these matrices we constructed the initial Fock matrix. The Mathematica software was used to program the matrix diagonalization process from the overlap and Hamiltonian core matrices and to make the recursion loop of the density matrix. Finally we obtained a value for the ground state energy of helium. Keywords: STO, Hartree Fock Roothaan approximation, spin orbital, ground state energy. Introduction The Hartree Fock method (Hartree, 1957) also known as Self Consistent Field (SCF) could be made with two types of spin orbital functions, the Slater Type Orbitals (STO) and the Gaussian Type Orbital (GTO), the election is made in function of the integrals of spin orbitals implicated. We used the STO to construct the wave function associated to the helium atom electrons. The application of the quantum mechanical theory was performed using a numerical method known as the finite element method, this gives the possibility to use the computer for programming the calculations quickly and efficiently (Becker, 1988; Brenner, 2008; Thomee, 2003; Reddy, 2004).
    [Show full text]